Math  /  Discrete

QuestionB. Q(1,1)Q(-1,1) C. x1(Q(x,x))\forall x \geq 1(Q(x, x)) D. x(¬Q(x,x))\exists x(\neg Q(x, x))
4. This question is about the rules of inference and proofs. (a) Fill in the following blanks (b) Consider the following argument. "If the book is large then it is heavy and it contains many topics. If the book is about sailing then it does not contain many topics. The book is about sailing. Therefore the book is not large." Use the rules of inference in table 4 to show that the above argument is a valid argument. To help you start, first name the simple statements:

Studdy Solution

STEP 1

1. We are dealing with logical expressions and rules of inference.
2. We need to fill in blanks for logical inference rules and validate an argument using these rules.
3. The rules of inference are assumed to be standard logical rules.

STEP 2

1. Fill in the blanks for the logical inference rules.
2. Name the simple statements in the argument.
3. Use the rules of inference to validate the argument.

STEP 3

Fill in the blanks for the logical inference rules:
1. Modus Ponens: pq,p p \to q, \, p therefore q q .
2. Hypothetical Syllogism: pq,qr,rs p \to q, \, q \to r, \, r \to s therefore ps p \to s .
3. Modus Tollens: p¬q p \land \neg q therefore ¬p \neg p .

STEP 4

Name the simple statements in the argument:
- Let L L be "The book is large." - Let H H be "The book is heavy." - Let M M be "The book contains many topics." - Let S S be "The book is about sailing."

STEP 5

Translate the argument into logical expressions:
1. If the book is large, then it is heavy and it contains many topics: L(HM) L \to (H \land M) .
2. If the book is about sailing, then it does not contain many topics: S¬M S \to \neg M .
3. The book is about sailing: S S .

STEP 6

Use the rules of inference to validate the argument:
1. From S¬M S \to \neg M and S S , by Modus Ponens, we get ¬M \neg M .
2. From L(HM) L \to (H \land M) and ¬M \neg M , by Modus Tollens, we get ¬L \neg L .

Therefore, the book is not large.
The argument is valid, and the conclusion is that the book is not large.

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